I printed the test image and have made the following observations, now of course, I am a total layman and this is just off the top of my head.

The background on the children, instead of white they printed more with a purplish white. The picture of the teardrop arch shot, the blue is more purple, the red/orange is perfectly aligned to the monitor color, the quadrant with the cd/remote is purpleish cast. The grey gradient is purple cast at 14, 16, 18.

The overall tone seems purple, so something is off, but I don't know what.

What do you suggest?

If you are sure your workflow is good: nozzle check print is good, Photoshop manages color, the correct profile selected, relative colorimetric with black point compensation (BPC shouldn't) matter for this purpose, and all color management turned off in the printer's dialog; then you probably have a poor quality profile.

However, the original print you posted is very dark. Is the test image also dark?